All of the 4th grade teachers and students from Gardner Bullis went on a field trip to an archaeology museum. Tickets were $$8.50$ each for teachers and $$2.50$ each for students, and the group paid $$42.00$ in total. A few weeks later, the same group visited a science museum where the tickets cost $$34.00$ each for teachers and $$11.00$ each for students, and the group paid $$178.00$ in total. Find the number of teachers and students on the field trips.
Answer: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${8.5x+2.5y = 42}$ ${34x+11y = 178}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-4$ ${-34x-10y = -168}$ ${34x+11y = 178}$ Add the top and bottom equations together. ${y = 10}$ Now that you know ${y = 10}$ , plug it back into $ {8.5x+2.5y = 42}$ to find $x$ ${8.5x + 2.5}{(10)}{= 42}$ $8.5x+25 = 42$ $8.5x = 17$ $x = \dfrac{17}{8.5}$ ${x = 2}$ You can also plug ${y = 10}$ into $ {34x+11y = 178}$ and get the same answer for $x$ ${34x + 11}{(10)}{= 178}$ ${x = 2}$ There were $2$ teachers and $10$ students on the field trips.